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Theorem psubclssatN 38812
Description: A closed projective subspace is a set of atoms. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
psubclssat.a 𝐴 = (Atomsβ€˜πΎ)
psubclssat.c 𝐢 = (PSubClβ€˜πΎ)
Assertion
Ref Expression
psubclssatN ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐢) β†’ 𝑋 βŠ† 𝐴)

Proof of Theorem psubclssatN
StepHypRef Expression
1 psubclssat.a . . 3 𝐴 = (Atomsβ€˜πΎ)
2 eqid 2733 . . 3 (βŠ₯π‘ƒβ€˜πΎ) = (βŠ₯π‘ƒβ€˜πΎ)
3 psubclssat.c . . 3 𝐢 = (PSubClβ€˜πΎ)
41, 2, 3psubcliN 38809 . 2 ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐢) β†’ (𝑋 βŠ† 𝐴 ∧ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) = 𝑋))
54simpld 496 1 ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐢) β†’ 𝑋 βŠ† 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   βŠ† wss 3949  β€˜cfv 6544  Atomscatm 38133  βŠ₯𝑃cpolN 38773  PSubClcpscN 38805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-psubclN 38806
This theorem is referenced by:  pmapidclN  38813  psubclinN  38819  paddatclN  38820  pclfinclN  38821  poml6N  38826  osumcllem3N  38829  osumcllem9N  38835  osumcllem11N  38837  osumclN  38838
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